Astronomy op o-v0j80e-1 chapter 24

be recorded by the telescope’s camera. These wide-field telescopes are not sensitive to faint sources, but ROTSE
showed that gamma-ray burst afterglows could sometimes be very bright.
Astronomers’ hopes were vindicated in March 2008, when an extremely bright gamma-ray burst occurred
and its light was captured by two wide-field camera systems in Chile: the Polish “Pi of the Sky” and the
Russian-Italian TORTORA [Telescopio Ottimizzato per la Ricerca dei Transienti Ottici Rapidi (Italian for Telescope
Optimized for the Research of Rapid Optical Transients)] (see Figure 23.21). According to the data taken by
these telescopes, for a period of about 30 seconds, the light from the gamma-ray burst was bright enough that
it could have been seen by the unaided eye had a person been looking in the right place at the right time.
Adding to our amazement, later observations by larger telescopes demonstrated that the burst occurred at a
distance of 8 billion light-years from Earth!
Figure 23.21 Gamma-Ray Burst Observed in March 2008. The extremely luminous afterglow of GRB 080319B was imaged by the Swift
Observatory in X-rays (left) and visible light/ultraviolet (right). (credit: modification of work by NASA/Swift/Stefan Immler, et al.)
To Beam or Not to Beam
The enormous distances to these events meant they had to have been astoundingly energetic to appear as
bright as they were across such an enormous distance. In fact, they required so much energy that it posed
a problem for gamma-ray burst models: if the source was radiating energy in all directions, then the energy
released in gamma rays alone during a bright burst (such as the 1999 or 2008 events) would have been
equivalent to the energy produced if the entire mass of a Sun-like star were suddenly converted into pure
radiation.
For a source to produce this much energy this quickly (in a burst) is a real challenge. Even if the star producing
the gamma-ray burst was much more massive than the Sun (as is probably the case), there is no known means
of converting so much mass into radiation within a matter of seconds. However, there is one way to reduce the
power required of the “mechanism” that makes gamma-ray bursts. So far, our discussion has assumed that the
source of the gamma rays gives off the same amount of energy in all directions, like an incandescent light bulb.
But as we discuss in Pulsars and the Discovery of Neutron Stars, not all sources of radiation in the universe
are like this. Some produce thin beams of radiation that are concentrated into only one or two directions. A laser
pointer and a lighthouse on the ocean are examples of such beamed sources on Earth (Figure 23.22). If, when a
burst occurs, the gamma rays come out in only one or two narrow beams, then our estimates of the luminosity
of the source can be reduced, and the bursts may be easier to explain. In that case, however, the beam has to
point toward Earth for us to be able to see the burst. This, in turn, would imply that for every burst we see from
Earth, there are probably many others that we never detect because their beams point in other directions.
Chapter 23 The Death of Stars 843

Figure 23.22 Burst That Is Beamed. This artist’s conception shows an illustration of one kind of gamma-ray burst. The collapse of the core of a
massive star into a black hole has produced two bright beams of light originating from the star’s poles, which an observer pointed along one of
these axes would see as a gamma-ray burst. The hot blue stars and gas clouds in the vicinity are meant to show that the event happened in an
active star-forming region. (credit: NASA/Swift/Mary Pat Hrybyk-Keith and John Jones)
Long-Duration Gamma-Ray Bursts: Exploding Stars
After identifying and following large numbers of gamma-ray bursts, astronomers began to piece together clues
about what kind of event is thought to be responsible for producing the gamma-ray burst. Or, rather, what kind
of events, because there are at least two distinct types of gamma-ray bursts. The two—like the different types of
supernovae—are produced in completely different ways.
Observationally, the crucial distinction is how long the burst lasts. Astronomers now divide gamma-ray bursts
into two categories: short-duration ones (defined as lasting less than 2 seconds, but typically a fraction of a
second) and long-duration ones (defined as lasting more than 2 seconds, but typically about a minute).
All of the examples we have discussed so far concern the long-duration gamma-ray bursts. These constitute
most of the gamma-ray bursts that our satellites detect, and they are also brighter and easier to pinpoint. Many
hundreds of long-duration gamma-ray bursts, and the properties of the galaxies in which they occurred, have
now been studied in detail. Long-duration gamma-ray bursts are universally observed to come from distant
galaxies that are still actively making stars. They are usually found to be located in regions of the galaxy with
strong star-formation activity (such as spiral arms). Recall that the more massive a star is, the less time it spends
in each stage of its life. This suggests that the bursts come from a young and short-lived, and therefore massive
type of star.
Furthermore, in several cases when a burst has occurred in a galaxy relatively close to Earth (within a few billion
light-years), it has been possible to search for a supernova at the same position—and in nearly all of these
cases, astronomers have found evidence of a supernova of type Ic going off. A type Ic is a particular type of
supernova, which we did not discuss in the earlier parts of this chapter; these are produced by a massive star
that has been stripped of its outer hydrogen layer. However, only a tiny fraction of type Ic supernovae produce
gamma-ray bursts.
Why would a massive star with its outer layers missing sometimes produce a gamma-ray burst at the same
time that it explodes as a supernova? The explanation astronomers have in mind for the extra energy is the
collapse of the star’s core to form a spinning, magnetic black hole or neutron star. Because the star corpse is
844 Chapter 23 The Death of Stars
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both magnetic and spinning rapidly, its sudden collapse is complex and can produce swirling jets of particles
and powerful beams of radiation—just like in a quasar or active galactic nucleus (objects you will learn about
Active Galaxies, Quasars, and Supermassive Black Holes), but on a much faster timescale. A small amount
of the infalling mass is ejected in a narrow beam, moving at speeds close to that of light. Collisions among the
particles in the beam can produce intense bursts of energy that we see as a gamma-ray burst.
Within a few minutes, the expanding blast from the fireball plows into the interstellar matter in the dying
star’s neighborhood. This matter might have been ejected from the star itself at earlier stages in its evolution.
Alternatively, it could be the gas out of which the massive star and its neighbors formed.
As the high-speed particles from the blast are slowed, they transfer their energy to the surrounding matter in
the form of a shock wave. That shocked material emits radiation at longer wavelengths. This accounts for the
afterglow of X-rays, visible light, and radio waves—the glow comes at longer and longer wavelengths as the
blast continues to lose energy.
Short-Duration Gamma-Ray Bursts: Colliding Stellar Corpses
What about the shorter gamma-ray bursts? The gamma-ray emission from these events lasts less than 2
seconds, and in some cases may last only milliseconds—an amazingly short time. Such a timescale is difficult
to achieve if they are produced in the same way as long-duration gamma-ray bursts, since the collapse of the
stellar interior onto the black hole should take at least a few seconds.
Astronomers looked fruitlessly for afterglows from short-duration gamma-ray bursts found by BeppoSAX and
other satellites. Evidently, the afterglows fade away too quickly. Fast-responding visible-light telescopes like
ROTSE were not helpful either: no matter how fast these telescopes responded, the bursts were not bright
enough at visible wavelengths to be detected by these small telescopes.
Once again, it took a new satellite to clear up the mystery. In this case, it was the Swift Gamma-Ray Burst Satellite,
launched in 2004 by a collaboration between NASA and the Italian and UK space agencies (Figure 23.23). The
design of Swift is similar to that of BeppoSAX. However, Swift is much more agile and flexible: after a gamma-
ray burst occurs, the X-ray and UV telescopes can be repointed automatically within a few minutes (rather
than a few hours). Thus, astronomers can observe the afterglow much earlier, when it is expected to be much
brighter. Furthermore, the X-ray telescope is far more sensitive and can provide positions that are 30 times
more precise than those provided by BeppoSAX, allowing bursts to be identified even without visible-light or
radio observations.

Figure 23.23 Artist’s Illustration of Swift. The US/UK/Italian spacecraft Swift contains on-board gamma-ray, X-ray, and ultraviolet detectors,
and has the ability to automatically reorient itself to a gamma-ray burst detected by the gamma-ray instrument. Since its launch in 2005, Swift
has detected and observed over a thousand bursts, including dozens of short-duration bursts. (credit: NASA, Spectrum Astro)
On May 9, 2005, Swift detected a flash of gamma rays lasting 0.13 seconds in duration, originating from the
constellation Coma Berenices. Remarkably, the galaxy at the X-ray position looked completely different from
any galaxy in which a long-duration burst had been seen to occur. The afterglow originated from the halo of
a giant elliptical galaxy 2.7 billion light-years away, with no signs of any young, massive stars in its spectrum.
Furthermore, no supernova was ever detected after the burst, despite extensive searching.
What could produce a burst less than a second long, originating from a region with no star formation? The
leading model involves the merger of two compact stellar corpses: two neutron stars, or perhaps a neutron star
and a black hole. Since many stars come in binary or multiple systems, it’s possible to have systems where two
such star corpses orbit one another. According to general relativity (which will be discussed in Black Holes and
Curved Spacetime), the orbits of a binary star system composed of such objects should slowly decay with time,
eventually (after millions or billions of years) causing the two objects to slam together in a violent but brief
explosion. Because the decay of the binary orbit is so slow, we would expect more of these mergers to occur in
old galaxies in which star formation has long since stopped.
While it was impossible to be sure of this model based on only a single event (it is possible this burst actually
came from a background galaxy and lined up with the giant elliptical only by chance), several dozen more short-
L I N K T O L E A R N I N G
To learn more about the merger of two neutron stars and how they can produce a burst that lasts less
than a second, check out this computer simulation (https://openstaxcollege.org/l/30comsimneustr)
by NASA.

duration gamma-ray bursts have since been located by Swift, many of which also originate from galaxies with
very low star-formation rates. This has given astronomers greater confidence that this model is the correct one.
Still, to be fully convinced, astronomers are searching for a “smoking gun” signature for the merger of two
ultra-dense stellar remnants.
Astronomers identified two observations that would provide more direct evidence. Theoretical calculations
indicate that when two neutron stars collide there will be a very special kind of explosion; neutrons stripped
from the neutron stars during the violent final phase of the merger will fuse together into heavy elements
and then release heat due to radioactivity, producing a short-lived but red supernova sometimes called a
kilonova. (The term is used because it is about a thousand times brighter than an ordinary nova, but not quite
as “super” as a traditional supernova.) Hubble observations of one short-duration gamma-ray burst in 2013
showed suggestive evidence of such a signature, but needed to be confirmed by future observations.
The second “smoking gun” is the detection of gravitational waves. As will be discussed in Black Holes and
Curved Spacetime, gravitational waves are ripples in the fabric of spacetime that general relativity predicts
should be produced by the acceleration of extremely massive and dense objects—such as two neutron stars or
black holes spiraling toward each other and colliding. The construction of instruments to detect gravitational
waves is very challenging technically, and gravitational wave astronomy became feasible only in 2015. The
first few detected gravitational wave events were produced by mergers of black holes. In 2017, however,
gravitational waves were observed from a source that was coincident in time and space with a gamma-ray
burst. The source consisted of two objects with the masses of neutron stars. A red supernova was also observed
at this location, and the ejected material was rich in heavy elements. This observation not only confirms the
theory of the origin of short gamma-ray bursts, but also is a spectacular demonstration of the validity of
Einstein’s theory of general relativity.
Probing the Universe with Gamma-Ray Bursts
The story of how astronomers came to explain the origin of the different kinds of bursts is a good example
of how the scientific process sometimes resembles good detective work. While the mystery of short-duration
gamma-ray bursts is still being unraveled, the focus of studies for long-duration gamma-ray bursts has begun
to change from understanding the origin of the bursts themselves (which is now fairly well-established) to using
them as tools to understand the broader universe.
The reason that long-duration gamma-ray bursts are useful has to do with their extreme luminosities, if only for
a short time. In fact, long-duration gamma-ray bursts are so bright that they could easily be seen at distances
that correspond to a few hundred million years after the expansion of the universe began, which is when
theorists think that the first generation of stars formed. Some theories predict that the first stars are likely to be
massive and complete their evolution in only a million years or so. If this turns out to be the case, then gamma-
ray bursts (which signal the death of some of these stars) may provide us with the best way of probing the
universe when stars and galaxies first began to form.
So far, the most distant gamma-ray burst found (on April 29, 2009) was in a galaxy with a redshift that
corresponds to a remarkable 13.2 billion light years—meaning it happened only 600 million years after the Big
Bang itself. This is comparable to the earliest and most distant galaxies found by the Hubble Space Telescope.
It is not quite old enough to expect that it formed from the first generation of stars, but its appearance at
this distance still gives us useful information about the production of stars in the early universe. Astronomers
continue to scan the skies, looking for even more distant events signaling the deaths of stars from even further
back in time.

Chandrasekhar limit
degenerate gas
millisecond pulsar
neutron star
nova
pulsar
type II supernova
CHAPTER 23 REVIEW
KEY TERMS
the upper limit to the mass of a white dwarf (equals 1.4 times the mass of the Sun)
a gas that resists further compression because no two electrons can be in the same place at
the same time doing the same thing (Pauli exclusion principle)
a pulsar that rotates so quickly that it can give off hundreds of pulses per second (and its
period is therefore measured in milliseconds)
a compact object of extremely high density composed almost entirely of neutrons
the cataclysmic explosion produced in a binary system, temporarily increasing its luminosity by hundreds
to thousands of times
a variable radio source of small physical size that emits very rapid radio pulses in very regular periods
that range from fractions of a second to several seconds; now understood to be a rotating, magnetic neutron
star that is energetic enough to produce a detectable beam of radiation and particles
a stellar explosion produced at the endpoint of the evolution of stars whose mass exceeds
roughly 10 times the mass of the Sun
SUMMARY
23.1 The Death of Low-Mass Stars
During the course of their evolution, stars shed their outer layers and lose a significant fraction of their initial
mass. Stars with masses of 8 MSun or less can lose enough mass to become white dwarfs, which have masses
less than the Chandrasekhar limit (about 1.4 MSun). The pressure exerted by degenerate electrons keeps white
dwarfs from contracting to still-smaller diameters. Eventually, white dwarfs cool off to become black dwarfs,
stellar remnants made mainly of carbon, oxygen, and neon.
23.2 Evolution of Massive Stars: An Explosive Finish
In a massive star, hydrogen fusion in the core is followed by several other fusion reactions involving heavier
elements. Just before it exhausts all sources of energy, a massive star has an iron core surrounded by shells
of silicon, sulfur, oxygen, neon, carbon, helium, and hydrogen. The fusion of iron requires energy (rather than
releasing it). If the mass of a star’s iron core exceeds the Chandrasekhar limit (but is less than 3 MSun), the core
collapses until its density exceeds that of an atomic nucleus, forming a neutron star with a typical diameter of
20 kilometers. The core rebounds and transfers energy outward, blowing off the outer layers of the star in a
type II supernova explosion.
23.3 Supernova Observations
A supernova occurs on average once every 25 to 100 years in the Milky Way Galaxy. Despite the odds, no
supernova in our Galaxy has been observed from Earth since the invention of the telescope. However, one
nearby supernova (SN 1987A) has been observed in a neighboring galaxy, the Large Magellanic Cloud. The star
that evolved to become SN 1987A began its life as a blue supergiant, evolved to become a red supergiant, and
returned to being a blue supergiant at the time it exploded. Studies of SN 1987A have detected neutrinos from
the core collapse and confirmed theoretical calculations of what happens during such explosions, including

the formation of elements beyond iron. Supernovae are a main source of high-energy cosmic rays and can be
dangerous for any living organisms in nearby star systems.
23.4 Pulsars and the Discovery of Neutron Stars
At least some supernovae leave behind a highly magnetic, rapidly rotating neutron star, which can be observed
as a pulsar if its beam of escaping particles and focused radiation is pointing toward us. Pulsars emit rapid
pulses of radiation at regular intervals; their periods are in the range of 0.001 to 10 seconds. The rotating
neutron star acts like a lighthouse, sweeping its beam in a circle and giving us a pulse of radiation when the
beam sweeps over Earth. As pulsars age, they lose energy, their rotations slow, and their periods increase.
23.5 The Evolution of Binary Star Systems
When a white dwarf or neutron star is a member of a close binary star system, its companion star can transfer
mass to it. Material falling gradually onto a white dwarf can explode in a sudden burst of fusion and make a
nova. If material falls rapidly onto a white dwarf, it can push it over the Chandrasekhar limit and cause it to
explode completely as a type Ia supernova. Another possible mechanism for a type Ia supernova is the merger
of two white dwarfs. Material falling onto a neutron star can cause powerful bursts of X-ray radiation. Transfer
of material and angular momentum can speed up the rotation of pulsars until their periods are just a few
thousandths of a second.
23.6 The Mystery of the Gamma-Ray Bursts
Gamma-ray bursts last from a fraction of a second to a few minutes. They come from all directions and are now
known to be associated with very distant objects. The energy is most likely beamed, and, for the ones we can
detect, Earth lies in the direction of the beam. Long-duration bursts (lasting more than a few seconds) come
from massive stars with their outer hydrogen layers missing that explode as supernovae. Short-duration bursts
are believed to be mergers of stellar corpses (neutron stars or black holes).
FOR FURTHER EXPLORATION
Articles
Death of Stars
Hillebrandt, W., et al. “How To Blow Up a Star.” Scientific American (October 2006): 42. On supernova
mechanisms.
Irion, R. “Pursuing the Most Extreme Stars.” Astronomy (January 1999): 48. On pulsars.
Kalirai, J. “New Light on Our Sun’s Fate.” Astronomy (February 2014): 44. What will happen to stars like our Sun
between the main sequence and the white dwarf stages.
Kirshner, R. “Supernova 1987A: The First Ten Years.” Sky & Telescope (February 1997): 35.
Maurer, S. “Taking the Pulse of Neutron Stars.” Sky & Telescope (August 2001): 32. Review of recent ideas and
observations of pulsars.
Zimmerman, R. “Into the Maelstrom.” Astronomy (November 1998): 44. About the Crab Nebula.
Gamma-Ray Bursts
Fox, D. & Racusin, J. “The Brightest Burst.” Sky & Telescope (January 2009): 34. Nice summary of the brightest
burst observed so far, and what we have learned from it.
Nadis, S. “Do Cosmic Flashes Reveal Secrets of the Infant Universe?” Astronomy (June 2008): 34. On different

types of gamma-ray bursts and what we can learn from them.
Naeye, R. “Dissecting the Bursts of Doom.” Sky & Telescope (August 2006): 30. Excellent review of gamma-ray
bursts—how we discovered them, what they might be, and what they can be used for in probing the universe.
Zimmerman, R. “Speed Matters.” Astronomy (May 2000): 36. On the quick-alert networks for finding afterglows.
Zimmerman, R. “Witness to Cosmic Collisions.” Astronomy (July 2006): 44. On the Swift mission and what it is
teaching astronomers about gamma-ray bursts.
Websites
Death of Stars
Crab Nebula: http://chandra.harvard.edu/xray_sources/crab/crab.html (http://chandra.harvard.edu/
xray_sources/crab/crab.html) . A short, colorfully written introduction to the history and science involving the
best-known supernova remant.
Introduction to Neutron Stars: https://www.astro.umd.edu/~miller/nstar.html
(https://www.astro.umd.edu/~miller/nstar.html) . Coleman Miller of the University of Maryland maintains
this site, which goes from easy to hard as you get into it, but it has lots of good information about corpses of
massive stars.
Introduction to Pulsars (by Maryam Hobbs at the Australia National Telescope Facility):
http://www.atnf.csiro.au/outreach/education/everyone/pulsars/index.html (http://www.atnf.csiro.au/
outreach/education/everyone/pulsars/index.html) .
Magnetars, Soft Gamma Repeaters, and Very Strong Magnetic Fields: http://solomon.as.utexas.edu/
magnetar.html (http://solomon.as.utexas.edu/magnetar.html) . Robert Duncan, one of the originators of
the idea of magnetars, assembled this site some years ago.
Gamma-Ray Bursts
Brief Intro to Gamma-Ray Bursts (from PBS’ Seeing in the Dark): http://www.pbs.org/seeinginthedark/
astronomy-topics/gamma-ray-bursts.html (http://www.pbs.org/seeinginthedark/astronomy-topics/
gamma-ray-bursts.html) .
Discovery of Gamma-ray Bursts: http://science.nasa.gov/science-news/science-at-nasa/1997/
ast19sep97_2/ (http://science.nasa.gov/science-news/science-at-nasa/1997/ast19sep97_2/) .
Gamma-Ray Bursts: Introduction to a Mystery (at NASA’s Imagine the Universe site):
http://imagine.gsfc.nasa.gov/docs/science/know_l1/bursts.html (http://imagine.gsfc.nasa.gov/docs/
science/know_l1/bursts.html) .
Introduction from the Swift Satellite Site: http://swift.sonoma.edu/about_swift/grbs.html
(http://swift.sonoma.edu/about_swift/grbs.html) .
Missions to Detect and Learn More about Gamma-ray Bursts:
• Fermi Space Telescope: http://fermi.gsfc.nasa.gov/public/ (http://fermi.gsfc.nasa.gov/public/) .
• INTEGRAL Spacecraft: http://www.esa.int/science/integral (http://www.esa.int/science/integral) .
• SWIFT Spacecraft: http://swift.sonoma.edu/ (http://swift.sonoma.edu/.) .

Videos
Death of Stars
BBC interview with Antony Hewish: http://www.bbc.co.uk/archive/scientists/10608.shtml
(http://www.bbc.co.uk/archive/scientists/10608.shtml) . (40:54).
Black Widow Pulsars: The Vengeful Corpses of Stars: https://www.youtube.com/watch?v=Fn-3G_N0hy4
(https://www.youtube.com/watch?v=Fn-3G_N0hy4) . A public talk in the Silicon Valley Astronomy Lecture
Series by Dr. Roger Romani (Stanford University) (1:01:47).
Hubblecast 64: It all ends with a bang!: http://www.spacetelescope.org/videos/hubblecast64a/
(http://www.spacetelescope.org/videos/hubblecast64a/) . HubbleCast Program introducing Supernovae
with Dr. Joe Liske (9:48).
Space Movie Reveals Shocking Secrets of the Crab Pulsar: http://hubblesite.org/newscenter/archive/
releases/2002/24/video/c/ (http://hubblesite.org/newscenter/archive/releases/2002/24/video/c/) . A
sequence of Hubble and Chandra Space Telescope images of the central regions of the Crab Nebula have been
assembled into a very brief movie accompanied by animation showing how the pulsar affects its environment;
it comes with some useful background material (40:06).
Gamma-Ray Bursts
Gamma-Ray Bursts: The Biggest Explosions Since the Big Bang!: https://www.youtube.com/
watch?v=ePo_EdgV764 (https://www.youtube.com/watch?v=ePo_EdgV764) . Edo Berge in a popular-level
lecture at Harvard (58:50).
Gamma-Ray Bursts: Flashes in the Sky: https://www.youtube.com/watch?v=23EhcAP3O8Q
(https://www.youtube.com/watch?v=23EhcAP3O8Q) . American Museum of Natural History Science Bulletin
on the Swift satellite (5:59).
Overview Animation of Gamma-Ray Burst: http://news.psu.edu/video/296729/2013/11/27/overview-
animation-gamma-ray-burst (http://news.psu.edu/video/296729/2013/11/27/overview-animation-
gamma-ray-burst) . Brief Animation of what causes a long-duration gamma-ray burst (0:55).
COLLABORATIVE GROUP ACTIVITIES
A. Someone in your group uses a large telescope to observe an expanding shell of gas. Discuss what
measurements you could make to determine whether you have discovered a planetary nebula or the
remnant of a supernova explosion.
B. The star Sirius (the brightest star in our northern skies) has a white-dwarf companion. Sirius has a mass of
about 2 MSun and is still on the main sequence, while its companion is already a star corpse. Remember that
a white dwarf can’t have a mass greater than 1.4 MSun. Assuming that the two stars formed at the same
time, your group should discuss how Sirius could have a white-dwarf companion. Hint: Was the initial mass
of the white-dwarf star larger or smaller than that of Sirius?
C. Discuss with your group what people today would do if a brilliant star suddenly became visible during the
daytime? What kind of fear and superstition might result from a supernova that was really bright in our
skies? Have your group invent some headlines that the tabloid newspapers and the less responsible web
news outlets would feature.

D. Suppose a supernova exploded only 40 light-years from Earth. Have your group discuss what effects there
may be on Earth when the radiation reaches us and later when the particles reach us. Would there be any
way to protect people from the supernova effects?
E. When pulsars were discovered, the astronomers involved with the discovery talked about finding “little
green men.” If you had been in their shoes, what tests would you have performed to see whether such a
pulsating source of radio waves was natural or the result of an alien intelligence? Today, several groups
around the world are actively searching for possible radio signals from intelligent civilizations. How might
you expect such signals to differ from pulsar signals?
F. Your little brother, who has not had the benefit of an astronomy course, reads about white dwarfs and
neutron stars in a magazine and decides it would be fun to go near them or even try to land on them. Is this
a good idea for future tourism? Have your group make a list of reasons it would not be safe for children (or
adults) to go near a white dwarf and a neutron star.
G. A lot of astronomers’ time and many instruments have been devoted to figuring out the nature of gamma-
ray bursts. Does your group share the excitement that astronomers feel about these mysterious high-
energy events? What are some reasons that people outside of astronomy might care about learning about
gamma-ray bursts?
EXERCISES
Review Questions
1. How does a white dwarf differ from a neutron star? How does each form? What keeps each from
collapsing under its own weight?
2. Describe the evolution of a star with a mass like that of the Sun, from the main-sequence phase of its
evolution until it becomes a white dwarf.
3. Describe the evolution of a massive star (say, 20 times the mass of the Sun) up to the point at which it
becomes a supernova. How does the evolution of a massive star differ from that of the Sun? Why?
4. How do the two types of supernovae discussed in this chapter differ? What kind of star gives rise to each
type?
5. A star begins its life with a mass of 5 MSun but ends its life as a white dwarf with a mass of 0.8 MSun. List
the stages in the star’s life during which it most likely lost some of the mass it started with. How did mass
loss occur in each stage?
6. If the formation of a neutron star leads to a supernova explosion, explain why only three of the hundreds
of known pulsars are found in supernova remnants.
7. How can the Crab Nebula shine with the energy of something like 100,000 Suns when the star that formed
the nebula exploded almost 1000 years ago? Who “pays the bills” for much of the radiation we see coming
from the nebula?
8. How is a nova different from a type Ia supernova? How does it differ from a type II supernova?
9. Apart from the masses, how are binary systems with a neutron star different from binary systems with a
white dwarf?

10. What observations from SN 1987A helped confirm theories about supernovae?
11. Describe the evolution of a white dwarf over time, in particular how the luminosity, temperature, and
radius change.
12. Describe the evolution of a pulsar over time, in particular how the rotation and pulse signal changes over
time.
13. How would a white dwarf that formed from a star that had an initial mass of 1 MSun be different from a
white dwarf that formed from a star that had an initial mass of 9 MSun?
14. What do astronomers think are the causes of longer-duration gamma-ray bursts and shorter-duration
gamma-ray bursts?
15. How did astronomers finally solve the mystery of what gamma-ray bursts were? What instruments were
required to find the solution?
Thought Questions
16. Arrange the following stars in order of their evolution:
A. A star with no nuclear reactions going on in the core, which is made primarily of carbon and oxygen.
B. A star of uniform composition from center to surface; it contains hydrogen but has no nuclear
reactions going on in the core.
C. A star that is fusing hydrogen to form helium in its core.
D. A star that is fusing helium to carbon in the core and hydrogen to helium in a shell around the core.
E. A star that has no nuclear reactions going on in the core but is fusing hydrogen to form helium in a
shell around the core.
17. Would you expect to find any white dwarfs in the Orion Nebula? (See The Birth of Stars and the
Discovery of Planets outside the Solar System to remind yourself of its characteristics.) Why or why not?
18. Suppose no stars more massive than about 2 MSun had ever formed. Would life as we know it have been
able to develop? Why or why not?
19. Would you be more likely to observe a type II supernova (the explosion of a massive star) in a globular
cluster or in an open cluster? Why?
20. Astronomers believe there are something like 100 million neutron stars in the Galaxy, yet we have only
found about 2000 pulsars in the Milky Way. Give several reasons these numbers are so different. Explain
each reason.
21. Would you expect to observe every supernova in our own Galaxy? Why or why not?
22. The Large Magellanic Cloud has about one-tenth the number of stars found in our own Galaxy. Suppose
the mix of high- and low-mass stars is exactly the same in both galaxies. Approximately how often does a
supernova occur in the Large Magellanic Cloud?
23. Look at the list of the nearest stars in Appendix I. Would you expect any of these to become supernovae?
Why or why not?
24. If most stars become white dwarfs at the ends of their lives and the formation of white dwarfs is
accompanied by the production of a planetary nebula, why are there more white dwarfs than planetary
nebulae in the Galaxy?

25. If a 3 and 8 MSun star formed together in a binary system, which star would:
A. Evolve off the main sequence first?
B. Form a carbon- and oxygen-rich white dwarf?
C. Be the location for a nova explosion?
26. You have discovered two star clusters. The first cluster contains mainly main-sequence stars, along with
some red giant stars and a few white dwarfs. The second cluster also contains mainly main-sequence
stars, along with some red giant stars, and a few neutron stars—but no white dwarf stars. What are the
relative ages of the clusters? How did you determine your answer?
27. A supernova remnant was recently discovered and found to be approximately 150 years old. Provide
possible reasons that this supernova explosion escaped detection.
28. Based upon the evolution of stars, place the following elements in order of least to most common in the
Galaxy: gold, carbon, neon. What aspects of stellar evolution formed the basis for how you ordered the
elements?
29. What observations or types of telescopes would you use to distinguish a binary system that includes a
main-sequence star and a white dwarf star from one containing a main-sequence star and a neutron star?
30. How would the spectra of a type II supernova be different from a type Ia supernova? Hint: Consider the
characteristics of the objects that are their source.
Figuring For Yourself
31. The ring around SN 1987A (Figure 23.12) initially became illuminated when energetic photons from the
supernova interacted with the material in the ring. The radius of the ring is approximately 0.75 light-year
from the supernova location. How long after the supernova did the ring become illuminated?
32. What is the acceleration of gravity (g) at the surface of the Sun? (See Appendix E for the Sun’s key
characteristics.) How much greater is this than g at the surface of Earth? Calculate what you would
weigh on the surface of the Sun. Your weight would be your Earth weight multiplied by the ratio of the
acceleration of gravity on the Sun to the acceleration of gravity on Earth. (Okay, we know that the Sun does
not have a solid surface to stand on and that you would be vaporized if you were at the Sun’s photosphere.
Humor us for the sake of doing these calculations.)
33. What is the escape velocity from the Sun? How much greater is it than the escape velocity from Earth?
34. What is the average density of the Sun? How does it compare to the average density of Earth?
35. Say that a particular white dwarf has the mass of the Sun but the radius of Earth. What is the acceleration
of gravity at the surface of the white dwarf? How much greater is this than g at the surface of Earth? What
would you weigh at the surface of the white dwarf (again granting us the dubious notion that you could
survive there)?
36. What is the escape velocity from the white dwarf in Exercise 23.35? How much greater is it than the escape
velocity from Earth?
37. What is the average density of the white dwarf in Exercise 23.35? How does it compare to the average
density of Earth?
38. Now take a neutron star that has twice the mass of the Sun but a radius of 10 km. What is the acceleration
of gravity at the surface of the neutron star? How much greater is this than g at the surface of Earth? What
would you weigh at the surface of the neutron star (provided you could somehow not become a puddle of
protoplasm)?

39. What is the escape velocity from the neutron star in Exercise 23.38? How much greater is it than the
escape velocity from Earth?
40. What is the average density of the neutron star in Exercise 23.38? How does it compare to the average
density of Earth?
41. One way to calculate the radius of a star is to use its luminosity and temperature and assume that the star
radiates approximately like a blackbody. Astronomers have measured the characteristics of central stars
of planetary nebulae and have found that a typical central star is 16 times as luminous and 20 times as
hot (about 110,000 K) as the Sun. Find the radius in terms of the Sun’s. How does this radius compare with
that of a typical white dwarf?
42. According to a model described in the text, a neutron star has a radius of about 10 km. Assume that
the pulses occur once per rotation. According to Einstein’s theory of relatively, nothing can move faster
than the speed of light. Check to make sure that this pulsar model does not violate relativity. Calculate
the rotation speed of the Crab Nebula pulsar at its equator, given its period of 0.033 s. (Remember that
distance equals velocity × time and that the circumference of a circle is given by 2πR).
43. Do the same calculations as in Exercise 23.42 but for a pulsar that rotates 1000 times per second.
44. If the Sun were replaced by a white dwarf with a surface temperature of 10,000 K and a radius equal to
Earth’s, how would its luminosity compare to that of the Sun?
45. A supernova can eject material at a velocity of 10,000 km/s. How long would it take a supernova remnant
to expand to a radius of 1 AU? How long would it take to expand to a radius of 1 light-years? Assume that
the expansion velocity remains constant and use the relationship: expansion time = distance
expansion velocity
.
46. A supernova remnant was observed in 2007 to be expanding at a velocity of 14,000 km/s and had a radius
of 6.5 light-years. Assuming a constant expansion velocity, in what year did this supernova occur?
47. The ring around SN 1987A (Figure 23.12) started interacting with material propelled by the shockwave
from the supernova beginning in 1997 (10 years after the explosion). The radius of the ring is
approximately 0.75 light-year from the supernova location. How fast is the supernova material moving,
assume a constant rate of motion in km/s?
48. Before the star that became SN 1987A exploded, it evolved from a red supergiant to a blue supergiant
while remaining at the same luminosity. As a red supergiant, its surface temperature would have been
approximately 4000 K, while as a blue supergiant, its surface temperature was 16,000 K. How much did the
radius change as it evolved from a red to a blue supergiant?
49. What is the radius of the progenitor star that became SN 1987A? Its luminosity was 100,000 times that of
the Sun, and it had a surface temperature of 16,000 K.
50. What is the acceleration of gravity at the surface of the star that became SN 1987A? How does this g
compare to that at the surface of Earth? The mass was 20 times that of the Sun and the radius was 41
times that of the Sun.
51. What was the escape velocity from the surface of the SN 1987A progenitor star? How much greater is it
than the escape velocity from Earth? The mass was 20 times that of the Sun and the radius was 41 times
that of the Sun.
52. What was the average density of the star that became SN 1987A? How does it compare to the average
density of Earth? The mass was 20 times that of the Sun and the radius was 41 times that of the Sun.
53. If the pulsar shown in Figure 23.16 is rotating 100 times per second, how many pulses would be detected
in one minute? The two beams are located along the pulsar’s equator, which is aligned with Earth.

Chapter Outline
24.1 Introducing General Relativity
24.2 Spacetime and Gravity
24.3 Tests of General Relativity
24.4 Time in General Relativity
24.5 Black Holes
24.6 Evidence for Black Holes
24.7 Gravitational Wave Astronomy
Thinking Ahead
For most of the twentieth century, black holes seemed the stuff of science fiction, portrayed either as monster
vacuum cleaners consuming all the matter around them or as tunnels from one universe to another. But
the truth about black holes is almost stranger than fiction. As we continue our voyage into the universe, we
will discover that black holes are the key to explaining many mysterious and remarkable objects—including
collapsed stars and the active centers of giant galaxies.
24.1 INTRODUCING GENERAL RELATIVITY
Learning Objectives
By the end of this section, you will be able to:
Figure 24.1 Stellar Mass Black Hole. On the left, a visible-light image shows a region of the sky in the constellation of Cygnus; the red box
marks the position of the X-ray source Cygnus X-1. It is an example of a black hole created when a massive star collapses at the end of its life.
Cygnus X-1 is in a binary star system, and the artist’s illustration on the right shows the black hole pulling material away from a massive blue
companion star. This material forms a disk (shown in red and orange) that rotates around the black hole before falling into it or being
redirected away from the black hole in the form of powerful jets. The material in the disk (before it falls into the black hole) is so hot that it
glows with X-rays, explaining why this object is an X-ray source. (credit left: modification of work by DSS; credit right: modification of work by
NASA/CXC/M.Weiss)
24
BLACK HOLES AND CURVED SPACETIME
Chapter 24 Black Holes and Curved Spacetime 857

Discuss some of the key ideas of the theory of general relativity
Recognize that one’s experiences of gravity and acceleration are interchangeable and indistinguishable
Distinguish between Newtonian ideas of gravity and Einsteinian ideas of gravity
Recognize why the theory of general relativity is necessary for understanding the nature of black holes
Most stars end their lives as white dwarfs or neutron stars. When a very massive star collapses at the end of
its life, however, not even the mutual repulsion between densely packed neutrons can support the core against
its own weight. If the remaining mass of the star’s core is more than about three times that of the Sun (MSun),
our theories predict that no known force can stop it from collapsing forever! Gravity simply overwhelms all other
forces and crushes the core until it occupies an infinitely small volume. A star in which this occurs may become
one of the strangest objects ever predicted by theory—a black hole.
To understand what a black hole is like and how it influences its surroundings, we need a theory that can
describe the action of gravity under such extreme circumstances. To date, our best theory of gravity is the
general theory of relativity, which was put forward in 1916 by Albert Einstein.
General relativity was one of the major intellectual achievements of the twentieth century; if it were music,
we would compare it to the great symphonies of Beethoven or Mahler. Until recently, however, scientists
had little need for a better theory of gravity; Isaac Newton’s ideas that led to his law of universal gravitation
(see Orbits and Gravity) are perfectly sufficient for most of the objects we deal with in everyday life. In the
past half century, however, general relativity has become more than just a beautiful idea; it is now essential
in understanding pulsars, quasars (which will be discussed in Active Galaxies, Quasars, and Supermassive
Black Holes), and many other astronomical objects and events, including the black holes we will discuss here.
We should perhaps mention that this is the point in an astronomy course when many students start to feel a
little nervous (and perhaps wish they had taken botany or some other earthbound course to satisfy the science
requirement). This is because in popular culture, Einstein has become a symbol for mathematical brilliance that
is simply beyond the reach of most people (Figure 24.2).
Figure 24.2 Albert Einstein (1879–1955). This famous scientist, seen here younger than in the usual photos, has become a symbol for high
intellect in popular culture. (credit: NASA)
So, when we wrote that the theory of general relativity was Einstein’s work, you may have worried just a bit,
convinced that anything Einstein did must be beyond your understanding. This popular view is unfortunate and
mistaken. Although the detailed calculations of general relativity do involve a good deal of higher mathematics,
the basic ideas are not difficult to understand (and are, in fact, almost poetic in the way they give us a new
perspective on the world). Moreover, general relativity goes beyond Newton’s famous “inverse-square” law of
858 Chapter 24 Black Holes and Curved Spacetime

gravity; it helps explain how matter interacts with other matter in space and time. This explanatory power is one
of the requirements that any successful scientific theory must meet.
The Principle of Equivalence
The fundamental insight that led to the formulation of the general theory of relativity starts with a very simple
thought: if you were able to jump off a high building and fall freely, you would not feel your own weight. In this
chapter, we will describe how Einstein built on this idea to reach sweeping conclusions about the very fabric of
space and time itself. He called it the “happiest thought of my life.”
Einstein himself pointed out an everyday example that illustrates this effect (see Figure 24.3). Notice how
your weight seems to be reduced in a high-speed elevator when it accelerates from a stop to a rapid descent.
Similarly, your weight seems to increase in an elevator that starts to move quickly upward. This effect is not just
a feeling you have: if you stood on a scale in such an elevator, you could measure your weight changing (you
can actually perform this experiment in some science museums).
Figure 24.3 Your Weight in an Elevator. In an elevator at rest, you feel your normal weight. In an elevator that accelerates as it descends, you
would feel lighter than normal. In an elevator that accelerates as it ascends, you would feel heavier than normal. If an evil villain cut the elevator
cable, you would feel weightless as you fell to your doom.
In a freely falling elevator, with no air friction, you would lose your weight altogether. We generally don’t like
to cut the cables holding elevators to try this experiment, but near-weightlessness can be achieved by taking
an airplane to high altitude and then dropping rapidly for a while. This is how NASA trains its astronauts for
the experience of free fall in space; the scenes of weightlessness in the 1995 movie Apollo 13 were filmed in
the same way. (Moviemakers have since devised other methods using underwater filming, wire stunts, and

computer graphics to create the appearance of weightlessness seen in such movies as Gravity and The Martian.)
Another way to state Einstein’s idea is this: suppose we have a spaceship that contains a windowless laboratory
equipped with all the tools needed to perform scientific experiments. Now, imagine that an astronomer wakes
up after a long night celebrating some scientific breakthrough and finds herself sealed into this laboratory. She
has no idea how it happened but notices that she is weightless. This could be because she and the laboratory
are far away from any source of gravity, and both are either at rest or moving at some steady speed through
space (in which case she has plenty of time to wake up). But it could also be because she and the laboratory
are falling freely toward a planet like Earth (in which case she might first want to check her distance from the
surface before making coffee).
What Einstein postulated is that there is no experiment she can perform inside the sealed laboratory to
determine whether she is floating in space or falling freely in a gravitational field.
[1]
As far as she is concerned,
the two situations are completely equivalent. This idea that free fall is indistinguishable from, and hence
equivalent to, zero gravity is called the equivalence principle.
Gravity or Acceleration?
Einstein’s simple idea has big consequences. Let’s begin by considering what happens if two foolhardy people
jump from opposite banks into a bottomless chasm (Figure 24.4). If we ignore air friction, then we can say that
while they freely fall, they both accelerate downward at the same rate and feel no external force acting on them.
They can throw a ball back and forth, always aiming it straight at each other, as if there were no gravity. The ball
falls at the same rate that they do, so it always remains in a line between them.
Watch how NASA uses a “weightless” environment (https://openstax.org/l/30NASAweightra) to help
train astronauts.
1 Strictly speaking, this is true only if the laboratory is infinitesimally small. Different locations in a real laboratory that is falling freely due to
gravity cannot all be at identical distances from the object(s) responsible for producing the gravitational force. In this case, objects in different
locations will experience slightly different accelerations. But this point does not invalidate the principle of equivalence that Einstein derived from
this line of thinking.

Figure 24.4 Free Fall. Two people play catch as they descend into a bottomless abyss. Since the people and ball all fall at the same speed, it
appears to them that they can play catch by throwing the ball in a straight line between them. Within their frame of reference, there appears to
be no gravity.
Such a game of catch is very different on the surface of Earth. Everyone who grows up feeling gravity knows
that a ball, once thrown, falls to the ground. Thus, in order to play catch with someone, you must aim the ball
upward so that it follows an arc—rising and then falling as it moves forward—until it is caught at the other end.
Now suppose we isolate our falling people and ball inside a large box that is falling with them. No one inside
the box is aware of any gravitational force. If they let go of the ball, it doesn’t fall to the bottom of the box or
anywhere else but merely stays there or moves in a straight line, depending on whether it is given any motion.
Astronauts in the International Space Station (ISS) that is orbiting Earth live in an environment just like that of
the people sealed in a freely falling box (Figure 24.5). The orbiting ISS is actually “falling” freely around Earth.
While in free fall, the astronauts live in a strange world where there seems to be no gravitational force. One
can give a wrench a shove, and it moves at constant speed across the orbiting laboratory. A pencil set in midair
remains there as if no force were acting on it.

Figure 24.5 Astronauts aboard the Space Shuttle. Shane Kimbrough and Sandra Magnus are shown aboard the Endeavour in 2008 with
various fruit floating freely. Because the shuttle is in free fall as it orbits Earth, everything—including astronauts—stays put or moves uniformly
relative to the walls of the spacecraft. This free-falling state produces a lack of apparent gravity inside the spacecraft. (credit: NASA)
Appearances are misleading, however. There is a force in this situation. Both the ISS and the astronauts
continually fall around Earth, pulled by its gravity. But since all fall together—shuttle, astronauts, wrench, and
pencil—inside the ISS all gravitational forces appear to be absent.
Thus, the orbiting ISS provides an excellent example of the principle of equivalence—how local effects of gravity
can be completely compensated by the right acceleration. To the astronauts, falling around Earth creates the
same effects as being far off in space, remote from all gravitational influences.
The Paths of Light and Matter
Einstein postulated that the equivalence principle is a fundamental fact of nature, and that there is no
experiment inside any spacecraft by which an astronaut can ever distinguish between being weightless in
remote space and being in free fall near a planet like Earth. This would apply to experiments done with
beams of light as well. But the minute we use light in our experiments, we are led to some very disturbing
conclusions—and it is these conclusions that lead us to general relativity and a new view of gravity.
It seems apparent to us, from everyday observations, that beams of light travel in straight lines. Imagine that a
spaceship is moving through empty space far from any gravity. Send a laser beam from the back of the ship to
the front, and it will travel in a nice straight line and land on the front wall exactly opposite the point from which
it left the rear wall. If the equivalence principle really applies universally, then this same experiment performed
in free fall around Earth should give us the same result.
Now imagine that the astronauts again shine a beam of light along the length of their ship. But, as shown in
Figure 24.6, this time the orbiting space station falls a bit between the time the light leaves the back wall and the
time it hits the front wall. (The amount of the fall is grossly exaggerated in Figure 24.6 to illustrate the effect.)
Therefore, if the beam of light follows a straight line but the ship’s path curves downward, then the light should
In the “weightless” environment of the International Space Station, moving takes very little effort. Watch
astronaut Karen Nyberg (https://openstax.org/l/30ISSzerogravid) demonstrate how she can propel
herself with the force of a single human hair.

strike the front wall at a point higher than the point from which it left.
Figure 24.6 Curved Light Path. In a spaceship moving to the left (in this figure) in its orbit about a planet, light is beamed from the rear, A,
toward the front, B. Meanwhile, the ship is falling out of its straight path (exaggerated here). We might therefore expect the light to strike at B′,
above the target in the ship. Instead, the light follows a curved path and strikes at C. In order for the principle of equivalence to be correct,
gravity must be able to curve the path of a light beam just as it curves the path of the spaceship.
However, this would violate the principle of equivalence—the two experiments would give different results. We
are thus faced with giving up one of our two assumptions. Either the principle of equivalence is not correct,
or light does not always travel in straight lines. Instead of dropping what probably seemed at the time like a
ridiculous idea, Einstein worked out what happens if light sometimes does not follow a straight path.
Let’s suppose the principle of equivalence is right. Then the light beam must arrive directly opposite the point
from which it started in the ship. The light, like the ball thrown back and forth, must fall with the ship that is in
orbit around Earth (see Figure 24.6). This would make its path curve downward, like the path of the ball, and
thus the light would hit the front wall exactly opposite the spot from which it came.
Thinking this over, you might well conclude that it doesn’t seem like such a big problem: why can’t light fall the
way balls do? But, as discussed in Radiation and Spectra, light is profoundly different from balls. Balls have
mass, while light does not.
Here is where Einstein’s intuition and genius allowed him to make a profound leap. He gave physical meaning
to the strange result of our thought experiment. Einstein suggested that the light curves down to meet the front
of the shuttle because Earth’s gravity actually bends the fabric of space and time. This radical idea—which we will
explain next—keeps the behavior of light the same in both empty space and free fall, but it changes some of
our most basic and cherished ideas about space and time. The reason we take Einstein’s suggestion seriously
is that, as we will see, experiments now clearly show his intuitive leap was correct.
24.2 SPACETIME AND GRAVITY
Learning Objectives
Describe Einstein’s view of gravity as the warping of spacetime in the presence of massive objects
Understand that Newton’s concept of the gravitational force between two massive objects and Einstein’s
concept of warped spacetime are different explanations for the same observed accelerations of one
massive object in the presence of another massive object
Is light actually bent from its straight-line path by the mass of Earth? How can light, which has no mass, be
affected by gravity? Einstein preferred to think that it is space and time that are affected by the presence of a
large mass; light beams, and everything else that travels through space and time, then find their paths affected.
Light always follows the shortest path—but that path may not always be straight. This idea is true for human

travel on the curved surface of planet Earth, as well. Say you want to fly from Chicago to Rome. Since an airplane
can’t go through the solid body of the Earth, the shortest distance is not a straight line but the arc of a great
circle.
Linkages: Mass, Space, and Time
To show what Einstein’s insight really means, let’s first consider how we locate an event in space and time. For
example, imagine you have to describe to worried school officials the fire that broke out in your room when your
roommate tried cooking shish kebabs in the fireplace. You explain that your dorm is at 6400 College Avenue, a
street that runs in the left-right direction on a map of your town; you are on the fifth floor, which tells where
you are in the up-down direction; and you are the sixth room back from the elevator, which tells where you are
in the forward-backward direction. Then you explain that the fire broke out at 6:23 p.m. (but was soon brought
under control), which specifies the event in time. Any event in the universe, whether nearby or far away, can be
pinpointed using the three dimensions of space and the one dimension of time.
Newton considered space and time to be completely independent, and that continued to be the accepted view
until the beginning of the twentieth century. But Einstein showed that there is an intimate connection between
space and time, and that only by considering the two together—in what we call spacetime—can we build up a
correct picture of the physical world. We examine spacetime a bit more closely in the next subsection.
The gist of Einstein’s general theory is that the presence of matter curves or warps the fabric of spacetime.
This curving of spacetime is identified with gravity. When something else—a beam of light, an electron, or
the starship Enterprise—enters such a region of distorted spacetime, its path will be different from what it
would have been in the absence of the matter. As American physicist John Wheeler summarized it: “Matter tells
spacetime how to curve; spacetime tells matter how to move.”
The amount of distortion in spacetime depends on the mass of material that is involved and on how
concentrated and compact it is. Terrestrial objects, such as the book you are reading, have far too little mass
to introduce any significant distortion. Newton’s view of gravity is just fine for building bridges, skyscrapers,
or amusement park rides. General relativity does, however, have some practical applications. The GPS (Global
Positioning System) in every smartphone can tell you where you are within 5 to 10 meters only because the
effects of general and special relativity on the GPS satellites in orbit around the Earth are taken into account.
Unlike a book or your roommate, stars produce measurable distortions in spacetime. A white dwarf, with its
stronger surface gravity, produces more distortion just above its surface than does a red giant with the same
mass. So, you see, we are eventually going to talk about collapsing stars again, but not before discussing
Einstein’s ideas (and the evidence for them) in more detail.
Spacetime Examples
How can we understand the distortion of spacetime by the presence of some (significant) amount of mass?
Let’s try the following analogy. You may have seen maps of New York City that squeeze the full three dimensions
of this towering metropolis onto a flat sheet of paper and still have enough information so tourists will not get
lost. Let’s do something similar with diagrams of spacetime.
Figure 24.7, for example, shows the progress of a motorist driving east on a stretch of road in Kansas where
the countryside is absolutely flat. Since our motorist is traveling only in the east-west direction and the terrain is
flat, we can ignore the other two dimensions of space. The amount of time elapsed since he left home is shown
on the y-axis, and the distance traveled eastward is shown on the x-axis. From A to B he drove at a uniform
speed; unfortunately, it was too fast a uniform speed and a police car spotted him. From B to C he stopped to
receive his ticket and made no progress through space, only through time. From C to D he drove more slowly
because the police car was behind him.

Figure 24.7 Spacetime Diagram. This diagram shows the progress of a motorist traveling east across the flat Kansas landscape. Distance
traveled is plotted along the horizontal axis. The time elapsed since the motorist left the starting point is plotted along the vertical axis.
Now let’s try illustrating the distortions of spacetime in two dimensions. In this case, we will (in our
imaginations) use a rubber sheet that can stretch or warp if we put objects on it.
Let’s imagine stretching our rubber sheet taut on four posts. To complete the analogy, we need something that
normally travels in a straight line (as light does). Suppose we have an extremely intelligent ant—a friend of the
comic book superhero Ant-Man, perhaps—that has been trained to walk in a straight line.
We begin with just the rubber sheet and the ant, simulating empty space with no mass in it. We put the ant on
one side of the sheet and it walks in a beautiful straight line over to the other side (Figure 24.8). We next put
a small grain of sand on the rubber sheet. The sand does distort the sheet a tiny bit, but this is not a distortion
that we or the ant can measure. If we send the ant so it goes close to, but not on top of, the sand grain, it has
little trouble continuing to walk in a straight line.
Now we grab something with a little more mass—say, a small pebble. It bends or distorts the sheet just a bit
around its position. If we send the ant into this region, it finds its path slightly altered by the distortion of the
sheet. The distortion is not large, but if we follow the ant’s path carefully, we notice it deviating slightly from a
straight line.
The effect gets more noticeable as we increase the mass of the object that we put on the sheet. Let’s say we
now use a massive paperweight. Such a heavy object distorts or warps the rubber sheet very effectively, putting
a good sag in it. From our point of view, we can see that the sheet near the paperweight is no longer straight.

Figure 24.8 Three-Dimensional Analogy for Spacetime. On a flat rubber sheet, a trained ant has no trouble walking in a straight line. When a
massive object creates a big depression in the sheet, the ant, which must walk where the sheet takes it, finds its path changed (warped)
dramatically.
Now let’s again send the ant on a journey that takes it close to, but not on top of, the paperweight. Far
away from the paperweight, the ant has no trouble doing its walk, which looks straight to us. As it nears the
paperweight, however, the ant is forced down into the sag. It must then climb up the other side before it can
return to walking on an undistorted part of the sheet. All this while, the ant is following the shortest path it can,
but through no fault of its own (after all, ants can’t fly, so it has to stay on the sheet) this path is curved by the
distortion of the sheet itself.
In the same way, according to Einstein’s theory, light always follows the shortest path through spacetime. But
the mass associated with large concentrations of matter distorts spacetime, and the shortest, most direct paths
are no longer straight lines, but curves.
How large does a mass have to be before we can measure a change in the path followed by light? In 1916, when
Einstein first proposed his theory, no distortion had been detected at the surface of Earth (so Earth might have
played the role of the grain of sand in our analogy). Something with a mass like our Sun’s was necessary to
detect the effect Einstein was describing (we will discuss how this effect was measured using the Sun in the next
section).
The paperweight in our analogy might be a white dwarf or a neutron star. The distortion of spacetime is greater
near the surfaces of these compact, massive objects than near the surface of the Sun. And when, to return to
the situation described at the beginning of the chapter, a star core with more than three times the mass of the
Sun collapses forever, the distortions of spacetime very close to it can become truly mind-boggling.
24.3 TESTS OF GENERAL RELATIVITY
Learning Objectives
Describe unusual motion of Mercury around the Sun and explain how general relativity explains the
observed behavior
Provide examples of evidence for light rays being bent by massive objects, as predicted by general
relativity’s theory about the warping of spacetime
What Einstein proposed was nothing less than a major revolution in our understanding of space and time.
It was a new theory of gravity, in which mass determines the curvature of spacetime and that curvature, in

turn, controls how objects move. Like all new ideas in science, no matter who advances them, Einstein’s theory
had to be tested by comparing its predictions against the experimental evidence. This was quite a challenge
because the effects of the new theory were apparent only when the mass was quite large. (For smaller masses,
it required measuring techniques that would not become available until decades later.)
When the distorting mass is small, the predictions of general relativity must agree with those resulting from
Newton’s law of universal gravitation, which, after all, has served us admirably in our technology and in guiding
space probes to the other planets. In familiar territory, therefore, the differences between the predictions of the
two models are subtle and difficult to detect. Nevertheless, Einstein was able to demonstrate one proof of his
theory that could be found in existing data and to suggest another one that would be tested just a few years
later.
The Motion of Mercury
Of the planets in our solar system, Mercury orbits closest to the Sun and is thus most affected by the distortion
of spacetime produced by the Sun’s mass. Einstein wondered if the distortion might produce a noticeable
difference in the motion of Mercury that was not predicted by Newton’s law. It turned out that the difference
was subtle, but it was definitely there. Most importantly, it had already been measured.
Mercury has a highly elliptical orbit, so that it is only about two-thirds as far from the Sun at perihelion as
it is at aphelion. (These terms were defined in the chapter on Orbits and Gravity.) The gravitational effects
(perturbations) of the other planets on Mercury produce a calculable advance of Mercury’s perihelion. What
this means is that each successive perihelion occurs in a slightly different direction as seen from the Sun (Figure
24.9).
Figure 24.9 Mercury’s Wobble. The major axis of the orbit of a planet, such as Mercury, rotates in space slightly because of various
perturbations. In Mercury’s case, the amount of rotation (or orbital precession) is a bit larger than can be accounted for by the gravitational
forces exerted by other planets; this difference is precisely explained by the general theory of relativity. Mercury, being the planet closest to the
Sun, has its orbit most affected by the warping of spacetime near the Sun. The change from orbit to orbit has been significantly exaggerated on
this diagram.
According to Newtonian gravitation, the gravitational forces exerted by the planets will cause Mercury’s
perihelion to advance by about 531 seconds of arc (arcsec) per century. In the nineteenth century, however,
it was observed that the actual advance is 574 arcsec per century. The discrepancy was first pointed out in
1859 by Urbain Le Verrier, the codiscoverer of Neptune. Just as discrepancies in the motion of Uranus allowed
astronomers to discover the presence of Neptune, so it was thought that the discrepancy in the motion of
Mercury could mean the presence of an undiscovered inner planet. Astronomers searched for this planet near
the Sun, even giving it a name: Vulcan, after the Roman god of fire. (The name would later be used for the home
planet of a famous character on a popular television show about future space travel.)

But no planet has ever been found nearer to the Sun than Mercury, and the discrepancy was still bothering
astronomers when Einstein was doing his calculations. General relativity, however, predicts that due to the
curvature of spacetime around the Sun, the perihelion of Mercury should advance slightly more than is
predicted by Newtonian gravity. The result is to make the major axis of Mercury’s orbit rotate slowly in space
because of the Sun’s gravity alone. The prediction of general relativity is that the direction of perihelion should
change by an additional 43 arcsec per century. This is remarkably close to the observed discrepancy, and it
gave Einstein a lot of confidence as he advanced his theory. The relativistic advance of perihelion was later also
observed in the orbits of several asteroids that come close to the Sun.
Deflection of Starlight
Einstein’s second test was something that had not been observed before and would thus provide an excellent
confirmation of his theory. Since spacetime is more curved in regions where the gravitational field is strong,
we would expect light passing very near the Sun to appear to follow a curved path (Figure 24.10), just like that
of the ant in our analogy. Einstein calculated from general relativity theory that starlight just grazing the Sun’s
surface should be deflected by an angle of 1.75 arcsec. Could such a deflection be observed?
Figure 24.10 Curvature of Light Paths near the Sun. Starlight passing near the Sun is deflected slightly by the “warping” of spacetime. (This
deflection of starlight is one small example of a phenomenon called gravitational lensing, which we’ll discuss in more detail in The Evolution
and Distribution of Galaxies.) Before passing by the Sun, the light from the star was traveling parallel to the bottom edge of the figure. When
it passed near the Sun, the path was altered slightly. When we see the light, we assume the light beam has been traveling in a straight path
throughout its journey, and so we measure the position of the star to be slightly different from its true position. If we were to observe the star
at another time, when the Sun is not in the way, we would measure its true position.
We encounter a small “technical problem” when we try to photograph starlight coming very close to the Sun:
the Sun is an outrageously bright source of starlight itself. But during a total solar eclipse, much of the Sun’s
light is blocked out, allowing the stars near the Sun to be photographed. In a paper published during World
War I, Einstein (writing in a German journal) suggested that photographic observations during an eclipse could
reveal the deflection of light passing near the Sun.
The technique involves taking a photograph of the stars six months prior to the eclipse and measuring the
position of all the stars accurately. Then the same stars are photographed during the eclipse. This is when the
starlight has to travel to us by skirting the Sun and moving through measurably warped spacetime. As seen
from Earth, the stars closest to the Sun will seem to be “out of place”—slightly away from their regular positions
as measured when the Sun is not nearby.
A single copy of that paper, passed through neutral Holland, reached the British astronomer Arthur S.
Eddington, who noted that the next suitable eclipse was on May 29, 1919. The British organized two expeditions
to observe it: one on the island of Príncipe, off the coast of West Africa, and the other in Sobral, in northern
Brazil. Despite some problems with the weather, both expeditions obtained successful photographs. The stars
seen near the Sun were indeed displaced, and to the accuracy of the measurements, which was about 20%, the
shifts were consistent with the predictions of general relativity. More modern experiments with radio waves

traveling close to the Sun have confirmed that the actual displacements are within 1% of what general relativity
predicts.
The confirmation of the theory by the eclipse expeditions in 1919 was a triumph that made Einstein a world
celebrity.
24.4 TIME IN GENERAL RELATIVITY
Learning Objectives
Describe how Einsteinian gravity slows clocks and can decrease a light wave’s frequency of oscillation
Recognize that the gravitational decrease in a light wave’s frequency is compensated by an increase in
the light wave’s wavelength—the so-called gravitational redshift—so that the light continues to travel at
constant speed
General relativity theory makes various predictions about the behavior of space and time. One of these
predictions, put in everyday terms, is that the stronger the gravity, the slower the pace of time. Such a statement
goes very much counter to our intuitive sense of time as a flow that we all share. Time has always seemed the
most democratic of concepts: all of us, regardless of wealth or status, appear to move together from the cradle
to the grave in the great current of time.
But Einstein argued that it only seems this way to us because all humans so far have lived and died in the
gravitational environment of Earth. We have had no chance to test the idea that the pace of time might
depend on the strength of gravity, because we have not experienced radically different gravities. Moreover, the
differences in the flow of time are extremely small until truly large masses are involved. Nevertheless, Einstein’s
prediction has now been tested, both on Earth and in space.
The Tests of Time
An ingenious experiment in 1959 used the most accurate atomic clock known to compare time measurements
on the ground floor and the top floor of the physics building at Harvard University. For a clock, the
experimenters used the frequency (the number of cycles per second) of gamma rays emitted by radioactive
cobalt. Einstein’s theory predicts that such a cobalt clock on the ground floor, being a bit closer to Earth’s
center of gravity, should run very slightly slower than the same clock on the top floor. This is precisely what
the experiments observed. Later, atomic clocks were taken up in high-flying aircraft and even on one of the
Gemini space flights. In each case, the clocks farther from Earth ran a bit faster. While in 1959 it didn’t matter
much if the clock at the top of the building ran faster than the clock in the basement, today that effect is highly
relevant. Every smartphone or device that synchronizes with a GPS must correct for this (as we will see in the
next section) since the clocks on satellites will run faster than clocks on Earth.
The effect is more pronounced if the gravity involved is the Sun’s and not Earth’s. If stronger gravity slows the
pace of time, then it will take longer for a light or radio wave that passes very near the edge of the Sun to
reach Earth than we would expect on the basis of Newton’s law of gravity. (It takes longer because spacetime
is curved in the vicinity of the Sun.) The smaller the distance between the ray of light and the edge of the Sun at
closest approach, the longer will be the delay in the arrival time.
In November 1976, when the two Viking spacecraft were operating on the surface of Mars, the planet went
behind the Sun as seen from Earth (Figure 24.11). Scientists had preprogrammed Viking to send a radio wave
toward Earth that would go extremely close to the outer regions of the Sun. According to general relativity,

there would be a delay because the radio wave would be passing through a region where time ran more slowly.
The experiment was able to confirm Einstein’s theory to within 0.1%.
Figure 24.11 Time Delays for Radio Waves near the Sun. Radio signals from the Viking lander on Mars were delayed when they passed near
the Sun, where spacetime is curved relatively strongly. In this picture, spacetime is pictured as a two-dimensional rubber sheet.
Gravitational Redshift
What does it mean to say that time runs more slowly? When light emerges from a region of strong gravity
where time slows down, the light experiences a change in its frequency and wavelength. To understand what
happens, let’s recall that a wave of light is a repeating phenomenon—crest follows crest with great regularity.
In this sense, each light wave is a little clock, keeping time with its wave cycle. If stronger gravity slows down the
pace of time (relative to an outside observer), then the rate at which crest follows crest must be correspondingly
slower—that is, the waves become less frequent.
To maintain constant light speed (the key postulate in Einstein’s theories of special and general relativity), the
lower frequency must be compensated by a longer wavelength. This kind of increase in wavelength (when
caused by the motion of the source) is what we called a redshift in Radiation and Spectra. Here, because it is
gravity and not motion that produces the longer wavelengths, we call the effect a gravitational redshift.
The advent of space-age technology made it possible to measure gravitational redshift with very high accuracy.
In the mid-1970s, a hydrogen maser, a device akin to a laser that produces a microwave radio signal at a
particular wavelength, was carried by a rocket to an altitude of 10,000 kilometers. Instruments on the ground
were used to compare the frequency of the signal emitted by the rocket-borne maser with that from a similar
maser on Earth. The experiment showed that the stronger gravitational field at Earth’s surface really did
slow the flow of time relative to that measured by the maser in the rocket. The observed effect matched the
predictions of general relativity to within a few parts in 100,000.
These are only a few examples of tests that have confirmed the predictions of general relativity. Today, general
relativity is accepted as our best description of gravity and is used by astronomers and physicists to understand
the behavior of the centers of galaxies, the beginning of the universe, and the subject with which we began this
chapter—the death of truly massive stars.
Relativity: A Practical Application
By now you may be asking: why should I be bothered with relativity? Can’t I live my life perfectly well without
it? The answer is you can’t. Every time a pilot lands an airplane or you use a GPS to determine where you are
on a drive or hike in the back country, you (or at least your GPS-enabled device) must take the effects of both
general and special relativity into account.

GPS relies on an array of 24 satellites orbiting the Earth, and at least 4 of them are visible from any spot on Earth.
Each satellite carries a precise atomic clock. Your GPS receiver detects the signals from those satellites that are
overhead and calculates your position based on the time that it has taken those signals to reach you. Suppose
you want to know where you are within 50 feet (GPS devices can actually do much better than this). Since it
takes only 50 billionths of a second for light to travel 50 feet, the clocks on the satellites must be synchronized
to at least this accuracy—and relativistic effects must therefore be taken into account.
The clocks on the satellites are orbiting Earth at a speed of 14,000 kilometers per hour and are moving much
faster than clocks on the surface of Earth. According to Einstein’s theory of relativity, the clocks on the satellites
are ticking more slowly than Earth-based clocks by about 7 millionths of a second per day. (We have not
discussed the special theory of relativity, which deals with changes when objects move very fast, so you’ll have
to take our word for this part.)
The orbits of the satellites are 20,000 kilometers above Earth, where gravity is about four times weaker than at
Earth’s surface. General relativity says that the orbiting clocks should tick about 45 millionths of a second faster
than they would on Earth. The net effect is that the time on a satellite clock advances by about 38 microseconds
per day. If these relativistic effects were not taken into account, navigational errors would start to add up and
positions would be off by about 7 miles in only a single day.
24.5 BLACK HOLES
Learning Objectives
Explain the event horizon surrounding a black hole
Discuss why the popular notion of black holes as great sucking monsters that can ingest material at great
distances from them is erroneous
Use the concept of warped spacetime near a black hole to track what happens to any object that might fall
into a black hole
Recognize why the concept of a singularity—with its infinite density and zero volume—presents major
challenges to our understanding of matter
Let’s now apply what we have learned about gravity and spacetime curvature to the issue we started with: the
collapsing core in a very massive star. We saw that if the core’s mass is greater than about 3 MSun, theory says
that nothing can stop the core from collapsing forever. We will examine this situation from two perspectives:
first from a pre-Einstein point of view, and then with the aid of general relativity.
Classical Collapse
Let’s begin with a thought experiment. We want to know what speeds are required to escape from the
gravitational pull of different objects. A rocket must be launched from the surface of Earth at a very high speed
if it is to escape the pull of Earth’s gravity. In fact, any object—rocket, ball, astronomy book—that is thrown into
the air with a velocity less than 11 kilometers per second will soon fall back to Earth’s surface. Only those objects
launched with a speed greater than this escape velocity can get away from Earth.
The escape velocity from the surface of the Sun is higher yet—618 kilometers per second. Now imagine that we
begin to compress the Sun, forcing it to shrink in diameter. Recall that the pull of gravity depends on both the
mass that is pulling you and your distance from the center of gravity of that mass. If the Sun is compressed, its
mass will remain the same, but the distance between a point on the Sun’s surface and the center will get smaller

and smaller. Thus, as we compress the star, the pull of gravity for an object on the shrinking surface will get
stronger and stronger (Figure 24.12).
Figure 24.12 Formation of a Black Hole. At left, an imaginary astronaut floats near the surface of a massive star-core about to collapse. As the
same mass falls into a smaller sphere, the gravity at its surface goes up, making it harder for anything to escape from the stellar surface.
Eventually the mass collapses into so small a sphere that the escape velocity exceeds the speed of light and nothing can get away. Note that the
size of the astronaut has been exaggerated. In the last picture, the astronaut is just outside the sphere we will call the event horizon and is
stretched and squeezed by the strong gravity.
When the shrinking Sun reaches the diameter of a neutron star (about 20 kilometers), the velocity required to
escape its gravitational pull will be about half the speed of light. Suppose we continue to compress the Sun to
a smaller and smaller diameter. (We saw this can’t happen to a star like our Sun in the real world because of
electron degeneracy, i.e., the mutual repulsion between tightly packed electrons; this is just a quick “thought
experiment” to get our bearings).
Ultimately, as the Sun shrinks, the escape velocity near the surface would exceed the speed of light. If the speed
you need to get away is faster than the fastest possible speed in the universe, then nothing, not even light, is
able to escape. An object with such large escape velocity emits no light, and anything that falls into it can never
return.
In modern terminology, we call an object from which light cannot escape a black hole, a name popularized by
the America scientist John Wheeler starting in the late 1960s (Figure 24.13). The idea that such objects might
exist is, however, not a new one. Cambridge professor and amateur astronomer John Michell wrote a paper in
1783 about the possibility that stars with escape velocities exceeding that of light might exist. And in 1796, the
French mathematician Pierre-Simon, marquis de Laplace, made similar calculations using Newton’s theory of
gravity; he called the resulting objects “dark bodies.”

Figure 24.13 John Wheeler (1911–2008). This brilliant physicist did much pioneering work in general relativity theory and popularized the term
black hole starting in the late 1960s. (credit: modification of work by Roy Bishop)
While these early calculations provided strong hints that something strange should be expected if very massive
objects collapse under their own gravity, we really need general relativity theory to give an adequate description
of what happens in such a situation.
Collapse with Relativity
General relativity tells us that gravity is really a curvature of spacetime. As gravity increases (as in the collapsing
Sun of our thought experiment), the curvature gets larger and larger. Eventually, if the Sun could shrink down
to a diameter of about 6 kilometers, only light beams sent out perpendicular to the surface would escape. All
others would fall back onto the star (Figure 24.14). If the Sun could then shrink just a little more, even that one
remaining light beam would no longer be able to escape.
Figure 24.14 Light Paths near a Massive Object. Suppose a person could stand on the surface of a normal star with a flashlight. The light
leaving the flashlight travels in a straight line no matter where the flashlight is pointed. Now consider what happens if the star collapses so that
it is just a little larger than a black hole. All the light paths, except the one straight up, curve back to the surface. When the star shrinks inside
the event horizon and becomes a black hole, even a beam directed straight up returns.
Keep in mind that gravity is not pulling on the light. The concentration of matter has curved spacetime, and
light (like the trained ant of our earlier example) is “doing its best” to go in a straight line, yet is now confronted
with a world in which straight lines that used to go outward have become curved paths that lead back in. The

collapsing star is a black hole in this view, because the very concept of “out” has no geometrical meaning. The
star has become trapped in its own little pocket of spacetime, from which there is no escape.
The star’s geometry cuts off communication with the rest of the universe at precisely the moment when, in
our earlier picture, the escape velocity becomes equal to the speed of light. The size of the star at this moment
defines a surface that we call the event horizon. It’s a wonderfully descriptive name: just as objects that sink
below our horizon cannot be seen on Earth, so anything happening inside the event horizon can no longer
interact with the rest of the universe.
Imagine a future spacecraft foolish enough to land on the surface of a massive star just as it begins to collapse
in the way we have been describing. Perhaps the captain is asleep at the gravity meter, and before the crew can
say “Albert Einstein,” they have collapsed with the star inside the event horizon. Frantically, they send an escape
pod straight outward. But paths outward twist around to become paths inward, and the pod turns around and
falls toward the center of the black hole. They send a radio message to their loved ones, bidding good-bye. But
radio waves, like light, must travel through spacetime, and curved spacetime allows nothing to get out. Their
final message remains unheard. Events inside the event horizon can never again affect events outside it.
The characteristics of an event horizon were first worked out by astronomer and mathematician Karl
Schwarzschild (Figure 24.15). A member of the German army in World War I, he died in 1916 of an illness he
contracted while doing artillery shell calculations on the Russian front. His paper on the theory of event horizons
was among the last things he finished as he was dying; it was the first exact solution to Einstein’s equations of
general relativity. The radius of the event horizon is called the Schwarzschild radius in his memory.
Figure 24.15 Karl Schwarzschild (1873–1916). This German scientist was the first to demonstrate mathematically that a black hole is possible
and to determine the size of a nonrotating black hole’s event horizon.
The event horizon is the boundary of the black hole; calculations show that it does not get smaller once the
whole star has collapsed inside it. It is the region that separates the things trapped inside it from the rest of
the universe. Anything coming from the outside is also trapped once it comes inside the event horizon. The
horizon’s size turns out to depend only on the mass inside it. If the Sun, with its mass of 1 MSun, were to become
a black hole (fortunately, it can’t—this is just a thought experiment), the Schwarzschild radius would be about
3 kilometers; thus, the entire black hole would be about one-third the size of a neutron star of that same mass.
Feed the black hole some mass, and the horizon will grow—but not very much. Doubling the mass will make
the black hole 6 kilometers in radius, still very tiny on the cosmic scale.
The event horizons of more massive black holes have larger radii. For example, if a globular cluster of 100,000

stars (solar masses) could collapse to a black hole, it would be 300,000 kilometers in radius, a little less than half
the radius of the Sun. If the entire Galaxy could collapse to a black hole, it would be only about 1012
kilometers
in radius—about a tenth of a light year. Smaller masses have correspondingly smaller horizons: for Earth to
become a black hole, it would have to be compressed to a radius of only 1 centimeter—less than the size of a
grape. A typical asteroid, if crushed to a small enough size to be a black hole, would have the dimensions of an
atomic nucleus.
E X A M P L E 2 4 . 1
The Milky Way’s Black Hole
The size of the event horizon of a black hole depends on the mass of the black hole. The greater the
mass, the larger the radius of the event horizon. General relativity calculations show that the formula for
the Schwarzschild radius (RS) of the event horizon is
RS = 2GM
c2
where c is the speed of light, G is the gravitational constant, and M is the mass of the black hole. Note
that in this formula, 2, G, and c are all constant; only the mass changes from black hole to black hole.
As we will see in the chapter on The Milky Way Galaxy, astronomers have traced the paths of several
stars near the center of our Galaxy and found that they seem to be orbiting an unseen object—dubbed
Sgr A* (pronounced “Sagittarius A-star”)—with a mass of about 4 million solar masses. What is the size
of its Schwarzschild radius?
Solution
We can substitute data for G, M, and c (from Appendix E) directly into the equation:
RS = 2GM
c2
=
2(6.67 × 10−11
N · m2
/kg2
)(4 × 106
)(1.99 × 1030
kg)
(3.00 × 108
m/s)2
= 1.18 × 1010
m
This distance is about one-fifth of the radius of Mercury’s orbit around the Sun, yet the object contains 4
million solar masses and cannot be seen with our largest telescopes. You can see why astronomers are
convinced this object is a black hole.
Check Your Learning
What would be the size of a black hole that contained only as much mass as a typical pickup truck (about
3000 kg)? (Note that something with so little mass could never actually form a black hole, but it’s
interesting to think about the result.)
Answer:
Substituting the data into our equation gives
RS = 2GM
c2
=
2(6.67 × 10−11
N · m2
/kg2
)(3000 kg)
(3.00 × 108
m/s)2
= 1.33 × 10−23
m.
For comparison, the size of a proton is usually considered to be about 8 × 10−16
m, which would be about

A Black Hole Myth
Much of the modern folklore about black holes is misleading. One idea you may have heard is that black holes
go about sucking things up with their gravity. Actually, it is only very close to a black hole that the strange effects
we have been discussing come into play. The gravitational attraction far away from a black hole is the same as
that of the star that collapsed to form it.
Remember that the gravity of any star some distance away acts as if all its mass were concentrated at a point
in the center, which we call the center of gravity. For real stars, we merely imagine that all mass is concentrated
there; for black holes, all the mass really is concentrated at a point in the center.
So, if you are a star or distant planet orbiting around a star that becomes a black hole, your orbit may not be
significantly affected by the collapse of the star (although it may be affected by any mass loss that precedes
the collapse). If, on the other hand, you venture close to the event horizon, it would be very hard for you to
resist the “pull” of the warped spacetime near the black hole. You have to get really close to the black hole to
experience any significant effect.
If another star or a spaceship were to pass one or two solar radii from a black hole, Newton’s laws would
be adequate to describe what would happen to it. Only very near the event horizon of a black hole is the
gravitation so strong that Newton’s laws break down. The black hole remnant of a massive star coming into our
neighborhood would be far, far safer to us than its earlier incarnation as a brilliant, hot star.
ten million times larger.
M A K I N G C O N N E C T I O N S
Gravity and Time Machines
Time machines are one of the favorite devices of science fiction. Such a device would allow you to move
through time at a different pace or in a different direction from everyone else. General relativity suggests
that it is possible, in theory, to construct a time machine using gravity that could take you into the future.
Let’s imagine a place where gravity is terribly strong, such as near a black hole. General relativity predicts
that the stronger the gravity, the slower the pace of time (as seen by a distant observer). So, imagine a
future astronaut, with a fast and strongly built spaceship, who volunteers to go on a mission to such a
high-gravity environment. The astronaut leaves in the year 2222, just after graduating from college at
age 22. She takes, let’s say, exactly 10 years to get to the black hole. Once there, she orbits some distance
from it, taking care not to get pulled in.
She is now in a high-gravity realm where time passes much more slowly than it does on Earth. This isn’t
just an effect on the mechanism of her clocks—time itself is running slowly. That means that every way
she has of measuring time will give the same slowed-down reading when compared to time passing on
Earth. Her heart will beat more slowly, her hair will grow more slowly, her antique wristwatch will tick
more slowly, and so on. She is not aware of this slowing down because all her readings of time, whether
made by her own bodily functions or with mechanical equipment, are measuring the
same—slower—time. Meanwhile, back on Earth, time passes as it always does.

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